How do you find the area bounded by the cardioid #r=1+cos(theta)#?

1 Answer
Mar 27, 2016

Answer:

#A=((cosx+4)sinx+3x)/4#
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Explanation:

Given: #r=1+cos(theta)#
Required: Area of cardioid?
Solution Strategy: Polar Coordinate Area Integral
#A= int_(theta_1)^(theta_2) 1/2r^2d(theta)# substitute for #r#
#A= 1/2int_(theta_1)^(theta_2) ( 1+cos(theta))^2d(theta)#
# =1/2[int (1+2costheta+cos^2theta) d(theta)]#
# = 1/2[theta + 2sintheta+ int cos^2theta d(theta)]#
#I_3=color(brown)(int cos^2theta d(theta))# apply reduction formula
#I_3=(n-1)/n int cos^(n-2)thetad(theta) +(cos^(n-1)thetasintheta)/n#
#I_3=color(brown)(1/2 theta +(costhetasintheta)/2)#
Putting it all together:

#A=((cosx+4)sinx+3x)/4#